Straight lines: y = mx + c
Drag two points and read the line through them: its gradient as rise over run, where it cuts the axes, and its equation with exact fractions.
Readouts
What's happening
The gradient m of a straight line is how far it goes up for every 1 across: pick any two points on it and divide the change in y (the rise) by the change in x (the run). Because the line is straight, every pair of points gives the same answer, which is why the gold triangle can sit anywhere on it. The constant c is where the line cuts the y-axis, so y = mx + c reads directly off the picture. Multiplying out the fractions gives the general form ax + by + c = 0 with whole numbers. Parallel lines share a gradient; perpendicular lines have gradients that multiply to −1, so one is the negative reciprocal of the other. The midpoint of AB averages the coordinates, and the length of AB comes from Pythagoras on the rise and the run.
GCSE Maths (AQA, Edexcel, OCR): equation of a straight line, gradient and intercept, parallel and perpendicular lines, midpoint and length of a line segment. A-Level Maths: coordinate geometry of straight lines.
Work through the numbers with Graphing Calculator and Fractions and Percentages.
Challenge
Predict first: the line through A(−2, −1) and B(4, 3). What is its gradient, and where does it cross the y-axis? Type the gradient as a decimal in the box, press Check, then move A and B there and read the gradient triangle.
The rise is 3 − (−1) = 4 and the run is 4 − (−2) = 6, so m = 4/6 = 2/3 ≈ 0.667. Then c = −1 − (2/3)(−2) = 1/3, so the line is y = (2/3)x + 1/3, or 2x − 3y + 1 = 0.
FAQ
- How do I find the equation of a line through two points?
- Find the gradient m = (y₂ − y₁)/(x₂ − x₁). Then put one point into y − y₁ = m(x − x₁) and rearrange to y = mx + c. For (−2, −1) and (4, 3): m = 4/6 = 2/3, and y + 1 = (2/3)(x + 2) gives y = (2/3)x + 1/3.
- How do I tell if two lines are perpendicular?
- Multiply their gradients. If the product is −1 they are perpendicular, so a line with gradient 2/3 is perpendicular to any line with gradient −3/2. Vertical and horizontal lines are the one exception: they are perpendicular but a vertical line has no gradient.
- What does c mean in y = mx + c?
- It is the y-intercept: the value of y where the line crosses the y-axis, because putting x = 0 into y = mx + c leaves y = c.
- How do I find the midpoint and length of AB?
- The midpoint averages the coordinates: ((x₁ + x₂)/2, (y₁ + y₂)/2). The length is √((x₂ − x₁)² + (y₂ − y₁)²), which is Pythagoras on the run and the rise.